In the present paper we consider the nonlinear superposition operator \(F\) in Banach spaces of sequences \(l_p\) \((1\le p\le \infty)\), generated by the function \(f(s, u) = d(s) + a^{ku} - 1\), with \(a > 1\) and \(k\in \mathbb{R}\setminus\{0\}\). We find out the Rhodius spectra \(\sigma_R(F)\) and the Neuberger spectra \(\sigma_N(F)\) of these operators, depending on the values of \(k\).

1Department of Internal Medicine, Public Hospital Travnik, Travnik, Bosnia and Herzegovina 2Department of Gynecology and Obstetrics, Public Hospital Travnik, Travnik, Bosnia and Herzegovina 3Department of Anesthesia and Resuscitation, Public Hospital Travnik, Travnik, Bosnia and Herzegovina 4Department of Neurology and Psychiatry, Public Hospital Travnik, Travnik, Bosnia and Herzegovina 5Public Institution Psychiatric Hospital of Canton

In this paper, we consider the topic from the theory of cosine operator functions in 2-dimensional real vector space, which is an interplay between functional analysis and matrix theory. For the various cases of a given real matrix A= [α , β; γ , δ] we find out the appropriate cosine operator function C(t)= [a(t), b(t); c(t), d(t)], (t \in R) in a real vector space R2 as the solutions of the Cauchy problem C''(t)=AC(t), C(0)=I, C'(0)=0.

. In this paper, we consider the topic from the theory of cosine operator functions in 2-dimensional real vector space, which is an interplay between functional analysis and matrix theory. For the various cases of a given real matrix A = , we find out the appropriate cosine operator function in a real vector space 2 , as the solutions of the

In this paper, we consider the nonlinear superposition operator F in lp spaces of sequences (1 ≤ p ≤ ∞), generated by the function f(s,u)=a(s) + arctan u or f(s,u) = a(s) - arctan u. We find out the Rhodius spectra σR(F) and the Neuberger spectra σN(F) of these operators and finally the radii of these spectra. The superposition operator generated by the function f(s,u) = a(s) ∓ arccot u appears to be a special case of above mentioned operator.

In this paper we consider the nonlinear superposition operator F in lp spaces of sequences, generated by the function f (s, u) = a (s) + u or f (s, u) = a (s) · u First we show that these operators are Fréchet differentiable. Then we find out the Neuberger spectra σN (F ) of these operators. We compare it with some other nonlinear spectra and indicate some possible applications.